When \(d=1\) , Conjecture REF  was proved by Héra, Keleti, and Máthé [1]} and when \(d=n-1,\)  Conjecture REF  was proved by D. Oberlin [2]}. In this paper, we will prove Conjecture REF  for the remaining values of \(d\) . As is typical for such problems, the conjecture will follow from the corresponding maximal function estimate, which we will state below. In what follows, \(\mathcal {L}_n\)  is the set of lines in \(\mathbb {R}^n\) , with the following metric: if \(\ell ,\ell ^{\prime }\in \mathcal {L}_n\) , then \(d(\ell ,\ell ^{\prime }) = |x-x^{\prime }| + |u\wedge u^{\prime }|\) . Here \(u\)  (resp \(u^{\prime }\) ) is a unit vector parallel to \(\ell \) ; \(x\)  (resp. \(x^{\prime }\) ) is the unique point on \(\ell \)  with \(x\perp u\) ; and \(|u\wedge u|\)  is the (unsigned) area of the parallelogram spanned by \(u\)  and \(u^{\prime }\) . A \(\delta \) -tube is the \(\delta \) -neighborhood of a unit line segment in \(\mathbb {R}^n\) , and if \(T\)  is a \(\delta \) -tube, then \(\ell (T)\)  is the line coaxial with \(T\) . Finally, if \(p\in [1,\infty ]\) , then \(p^{\prime }\)  denotes the conjugate Lebesgue exponent.
